Paths in Weyl chambers and random matrices
The largest eigenvalue of the random matrix of the Gaussian unitary ensemble of order \(d\) has the same distribution as \[ \max_{1\geq t_1 \geq \cdots \geq t_{d-1} \geq 0} [W_1(1)-W_1(t_1)+W_2(t_1)-W_2(t_2)+ \cdots +W_d(t_{d-1})], \] where \(W_1\), \(\dots\), \(W_d\) are independent standard real Brownian motions. This result was proved in a path representation by \textit{Yu. Baryshnikov} [Probab. Theory Related Fields 119, 256--274 (2001; Zbl 0980.60042)] and \textit{J. Gravner, C. A. Tracy} and \textit{H. Widom} [J. Stat. Phys. 102, 1085--1132 (2001; Zbl 0989.82030)]. In this paper, a similar formula is given for all eigenvalues, using a geometric approach. A continuous projection is introduced associating a path \({\mathcal T}w\) in the closure of the Weyl chamber \({\mathbf a}_+\) to a path \(w\) in an Abelian subspace \({\mathbf a}\). This path transformation was used in connection with a representation theory of complex group already by \textit{O. Mathieu} [Astérisque 237, 209--224 (1996; Zbl 0886.17009)] and it also appeared in the paper of \textit{J. W. Pitman} [Adv. Appl. Probab. 7, 511--526 (1975; Zbl 0332.60055)], where \(w\) is the usual Brownian motion and \({\mathcal T}w\) was then a \(3\)-dimensional Bessel process. The presented results can be seen as a generalization of Pitman's result to Weyl chambers. If \({\mathbf a}={\mathbb R}^{d}\), \({\mathbf a}_+=\{(x_1, \cdots, x_d); x_1 > x_2 > \cdots > x_d\}\) and \(W\) is the Euclidean Brownian motion on the space \({\mathbf a}\) then \({\mathcal T}W\) is the process of the eigenvalues of the Dyson Brownian motion on the set of Hermitian matrices and the transformation \(({\mathcal T}W)(1)\) has the distribution as the eigenvalues of the Gaussian unitary ensemble.
scientific article; zbMATH DE number 1903693